CHARACTER SUMS AND THE SERIES l(1,X) WITH APPLICATIONS TO REAL QUADRATIC FIELDS

摘要

In this article ,let k≡0 or 1 (mod 4) be a fundamental discriminant, and let x(n) be the real even primitive character modulo k. The series L(1,x)=∑~∞ _n=1 x(n) can be divided into groups of k consecutive terms. Let vbe any nonnegative integer, j an integer, 0 <= j ≤ k-1, and left T(u,j,x)= ∑~j+k_n=j+1 x(vk+n)/vk+n Then L(1,x)=∑~∞_v=0 T(u,,x)=∑ ~j_n=1 x(v)/+∑~∞_v=0T(u,j,x). we will derive algorithm and formula for calculating the class number of a real quadratic field. In section 4,we will attempt to make a connection between two conjectures on real quadratic fields and the sign of T(0,20,x).